Particle size statistics

Transcription

1 Impactors and Particle Size Distribution (2) National Institute for Occupational Safety and Health Division of Respiratory Disease Studies Field Studies Branch Ju-Hyeon Park, Sc.D., M.P.H., C.I.H. Particle size statistics

2 Aerosol measurement and size distribution Increasin Complexity Concentration Diameter. Concentration Particle mass, surface area or number per unit volume 2. Size distribution Concentation versus particle size Aerosol particle sizin Sample Assin to size bin.4 µm < d <.7 µm Size particles Bins Particles are assined to bins accordin to particle diameter 2

3 Conversion of a discrete particle size distribution to a continuous distribution.2 Discrete Distribution Number Number concentration.5..5 Number concentation is proportional to the heiht of each bar Particle size bins Particle Diameter /.4 Number Concentation / Bin Continuous Distribution Number concentation is proportional to the area of each bar Particle Diameter / 3

4 Number Concentation / Bin dn/d( d ) Continuous Distribution The smooth continuous distribution is obtained by joinin the bin mid-points Particle Diameter / Continuous raph Y-axis Differential particle concentration (or number) Particle number normalized by rane of particle diameter of the interval (or bin) dn/dd X-axis Particle diameter rane of the interval dd Interated area under the curve=total # of particle dn dd = n dd 4

5 Properties of particle size distribution Asymmetrical (or skewed) distribution Lon tail to the riht Lare number (fraction) of small particles Small number (fraction) of lare particles Lare rane of particle size Several orders of manitude in particle diameter No neative particle size Mode < Median < Mean Geometric mean (d or GM) Lo d =(Σn i lod i )/N d =exp{(σn i lod i )/N} Arithmetic mean and GM Arithmetic mean d a =(Σn i d i )/N = (summation of all areas of the bars) / (total number of particles) Geometric mean Lo d =(Σn i lod i )/N d =exp{(σn i lod i )/N} 5

6 Number Concentation / Bin dn/d( d ) Continuous Distribution The smooth continuous distribution is obtained by joinin the bin mid-points Particle Diameter / Asymmetric (skewed) distribution.8 Mode Modal Diameter Lonormalormal Size Distribution Normalized distribution (n=) Area inder curve = Arithmetic mean dn/d(d) Note that when plotted as dn/d(d), the modal and count median diameters of a lonormal distribution are different Geometric mean (Median) Distribution skewed to smaller particle diameters CMD 6

7 Normal distribution.8 Normalized distribution (n=) Area inder curve = d p Normal Size Distribution Mean = Mode = Median dn/d(d) % of all particles are between ± σ Distribution characterized by n, and σ d p -σ 2 +σ df = Normal distribution n σ 2π e ( d p d p ) 2 2σ 2 dd p dp: arithmetic mean particle diameter σ: standard deviation ddp: particle diameter interval df: frequency of occurrence of particles of diameter dp 7

8 Lo transformation of continuous raph Y-axis Differential particle concentration (or number) Particle number normalized by rane of lo-transformed particle diameter of the interval (or bin) dn/dlo(d) X-axis Rane of lo-transformed particle diameter of the interval dlo(d) Interated area under the curve=total # of particle {dn/dlo(d)}dlo(d) Lonormal distribution with arithmetic scale dn/d(d) Modal Diameter Lonormalormal Size Distribution Normalized distribution (n=) Area inder curve = Count median diameter (CMD)=GM Note that when plotted as dn/d(d), the modal and count median diameters of a lonormal distribution are different Distribution skewed to smaller particle diameters CMD 8

9 Mathematical function of lonormal distribution df = n 2π Lo(σ ) e ( Lo( d p ) Lo( CMD ) 2 ( ) 2Lo ( σ ) 2 dlo d p df = n σ 2π e ( d p d p ) 2 2σ 2 dd p Two ways of lo-transformed raph Transform the oriinal particle size data usin loarithm, and then plot them on normal arithmetic scale of the raph To calculate all statistics mathematically Exponentiate lo-transformed statistics Transform x-axis scale of the raph, and then plot the oriinal particle size data on it Do not transform the data Only chane the scale of the raph 9

10 Lo normal distribution (first approach).8 Normalized distribution (n=) Area inder curve = d p Normal Size Distribution Mean = Mode = Median dn/d(d) % of all particles are between ± σ Distribution characterized by n, and σ d p -σ 2 +σ Lonormal distribution with lo scale.5 Normalized distribution (n=) Area inder curve = Lonormal Size Distribution dn/dlo(d).5 Count Median Diameter (CMD) 6% of all particles are less than CMD/σ 5% of all particles are less than CMD 68% of all particles are between CMD / σ and CMD σ 84% of all particles are less than CMD*σ Distribution characterized by n, CMD and σ.. CMD/σ CMD σ

11 Cumulative size distribution.8 Normalized distribution (n=) Area inder curve = 84% Lonormal Size Distribution Fraction of particles smaller than d % CMD 6%. CMD/ σ CMD σ Probability scale of lonormal distribution Percent of particles less than a iven particle diameter <CMD/(2σ )- 5% <CMD/σ - 6 % <CMD (median)- 5% <CMD*σ - 84% <CMD*(2σ )- 95% Any pattern?? Symmetry of probability

12 Lonormal distribution with lo scale.5 Normalized distribution (n=) Area inder curve = Lonormal Size Distribution Count Median Diameter (CMD) 5% of all particles are less than CMD dn/dlo(d) 68% of all particles are between CMD / σ and CMD σ.5 6% of all particles are less than CMD/σ 84% of all particles are less than CMD*σ Distribution characterized by n, CMD and σ.. CMD/σ CMD σ Cumulative plot to lo-probability plot.8 Normalized distribution (n=) Area inder curve = 84% Lonormal Size Distribution Fraction of particles smaller than d % CMD Switch and chane to probability scale 6%. CMD/ σ CMD σ 2

13 Lo scale (particle size: µm) Lo-probability paper Probability scale (probit) Lo-probability plot of lonormal distribution 6% 5% 84% CMD σ CMD CMD/σ 2 3 /2 = σ = 2/ % Distribution less than diameter d (probit) Lonormal Size Distribution 3

14 Count median diameter and σ CMD (count median diameter) 5% of all particles are less than CMD Geometric standard deviation CMD*σ /CMD or CMD/(CMD/σ ).5 CMD, SMD, and MMD Normalized distributions Number weihted ( X=n) Surface-area weihted (X=s) Mass weihted (X=m) dx/dlo(d).5. CMD SMD MMD 4

15 Lo-probability plots for count, mass, and surface area lonormal distribution 6% 5% 84% Number weihted Surface-area weihted Mass weihted. Plots are parallel - σ (iven by radient) is the same for each weihtin % Distribution less than diameter d (probit) Cascade impactor data reduction Stae # Size Rane (µm) d 5 (µm) Initial Mass (m) Final Mass (m) Net Mass (m) Mass Fraction (%) Cumm. Mass Fraction (%) > Back filter Total

16 An example usin count data Data William Hinds, Aerosol Technoloy, 2 nd (999) 6

17 Plottin upper size rane vs. fraction/µm..8 Fraction/µm Particle size (µm) Lo scale of particle diameter (lonormal distribution)..8 Fraction/µm Particle size (µm) 7

18 Cumulative lonormal plot (cum vs. size) 8 Cumulative fraction (%) Particle size (µm) Lo-probability plot of count data Particle size in lo scale (µm) % 5% 84% Cumulative fraction in probability scale/µm 8

19 Take-home practice Stae cut-point (diameter d) Mass concentration Cumulative mass concentration reater than diameter d (µm) (m/m 3 ) (m/m 3 ) Sum (m/m 3 ) % Cumulative mass concentration reater than diameter % Cumulative mass concentration less than diameter. Fill out the blank columns. 2. Find out MMD (Mass Median Diameter) and Geometric standard deviation (GSD). 9